Phase-based fast 3D high-resolution quantitative T2 MRI in 7 T human brain imaging

Magnetic resonance imaging (MRI) is a powerful and versatile technique that offers a range of physiological, diagnostic, structural, and functional measurements. One of the most widely used basic contrasts in MRI diagnostics is transverse relaxation time (T2)-weighted imaging, but it provides only qualitative information. Realizing quantitative high-resolution T2 mapping is imperative for the development of personalized medicine, as it can enable the characterization of diseases progression. While ultra-high-field (≥ 7 T) MRI offers the means to gain new insights by increasing the spatial resolution, implementing fast quantitative T2 mapping cannot be achieved without overcoming the increased power deposition and radio frequency (RF) field inhomogeneity at ultra-high-fields. A recent study has demonstrated a new phase-based T2 mapping approach based on fast steady-state acquisitions. We extend this new approach to ultra-high field MRI, achieving quantitative high-resolution 3D T2 mapping at 7 T while addressing RF field inhomogeneity and utilizing low flip angle pulses; overcoming two main ultra-high field challenges. The method is based on controlling the coherent transverse magnetization in a steady-state gradient echo acquisition; achieved by utilizing low flip angles, a specific phase increment for the RF pulses, and short repetition times. This approach simultaneously extracts both T2 and RF field maps from the phase of the signal. Prior to in vivo experiments, the method was assessed using a 3D head-shaped phantom that was designed to model the RF field distribution in the brain. Our approach delivers fast 3D whole brain images with submillimeter resolution without requiring special hardware, such as multi-channel transmit coil, thus promoting high usability of the ultra-high field MRI in clinical practice.

. Schematics of the steady state method for T 2 and RF field estimation-its design and verification. Starting from simulations, through assessment of the estimation algorithm, via a 3D head-shaped brain-like phantom, to human imaging. Left-a design of a steady-state configuration based on Bloch simulations that provides θ(T 2 ,α) for specific φ inc, which was thereafter utilized to generate T 2 and α in the 2D space (θ 1 , θ 2 ). The new space allows to extract T 2 and α from θ 1 and θ 2 . Center-the estimation algorithm was assessed via simulations and brain-like phantom measurements. In these measurements, a realistic signal S was acquired, providing |S| and ∠S , from which the T 2 and α (or B 1 distribution) were estimated. Right-human imaging at 7 T MRI provided high-resolution whole-brain T 2 maps, while coping with the B 1 distribution.  31,32 . They showed that an RF pulse train with a quadratic phase φ RF (n) = φ inc •(n 2 + n)/2 for the n-th pulse-using an appropriate φ inc value in conjunction with a spoiling gradient-can achieve incoherent transverse magnetization, an effective spoiling better than simple gradient spoiling (φ inc = 0 case). This is commonly called RF spoiling. Recent work by Wang 27 at 3 T provided another keystone, in which the authors showed that small φ inc values have the opposite effect; they introduce coherent transverse magnetization, where the phase of the signal possess a strong dependence on T 2 (Fig. 2a). Figure 2a also shows the dependence of the phase of the signal on the excitation flip angle α. The α dependence curves have an extremum in the vicinity of 15°, i.e., the actual flip angle in that vicinity has a small effect on the phase. As the flip angle in the 3 T implementation was assumed to be given by the scan (due to relatively homogeneous RF field distribution), T 2 values could be extracted solely from the phase of the signal. In our study, however, the combined (T 2 , α) dependence of the signal's phase θ was exploited to cope with the RF field inhomogeneity at ultra-high field MRI. Neglecting, for now, the small T 1 dependence of the phase-for T 1 values relevant to brain tissues at 7 T, see Fig. 2a-the phase θ of the signal depends on the T 2 at the voxel and on the actual flip angle α there. This α is the target flip angle of the scan α scan scaled by the RF field ratio at each voxel: α = α scan •RF ratio , where RF ratio is the normalized RF field distribution. As the phase θ(T 2 , α) (see Fig. 2a) is not a one-to-one map of (T 2 , α) to θ, at least two measurements, θ 1 and θ 2 , are needed; thus defining a 2D space (θ 1 , θ 2 ). To extract T 2 and α from θ(T 2 ,α), we need a convenient 2D space to represent T 2 and α in each voxel . Based on the Bloch simulations, such a 2D space can be generated by two scans with two flip angles, α scan1 and α scan2 = R FA •α scan1 (R FA is a user set multiplication factor; for example, R FA = 2). Furthermore, we found that varying φ inc between the two scans-one scan with (φ inc1, α scan1 ) and a second with (φ inc2, α scan2 = R FA • α scan1 )-provides greater flexibility in controlling the 2D (θ 1 , θ 2 ) space and its mapping to (T 2 , α). Figure 2b shows that different combinations of phase increment and flip angle pairs can be useful to adjust the range of viable flip angles and the T 2 of interest.
Variability and bias evaluation + SAR considerations. We examined the variability and bias of the method in the range of flip angles relevant for brain imaging. To do so, noise was added to the simulated signal and the variability and bias of the method were examined as a function of T 2 and α. The noise in the simulations was calibrated so the resulting synthetic signal to noise ratio (SNR) matched the measured SNR in agar tubes for the same α and T 2 , where the agar T 2 was in a range matching white matter (WM) and gray matter (GM) at 7T 18 . The signal dependence on flip angle and phase increment showed that the phase of the signal is high for low φ inc (φ inc < 10°) (Fig. S1). It can be seen that the combination (φ inc1 = 3°, α scan1 ) and (φ inc2 = 1.5°, α scan2 = 2α scan1 ) provides a lower variability (i.e., lower std(T 2 est.
) − T 2 true |) for a larger range of flip angles (Fig. S2). We also examined three criteria (Fig. S3): the average estimation variability for 30 < T 2 < 50 ms and 5° < α < 17°, and both the minimal and maximal flip angles that provide std(T 2 est.
) and supports a flip angle range of 3.7-35° (in which std(T 2 est. ) < 5 ms). Figure S4 shows three additional aspects that were included to establish the final configuration, including the repetition time (TR), the R FA in a realistic experiment and reduction of the cerebrospinal fluid (CSF) signal. Although the combination (φ inc1 = 3°, α scan1 ) and (φ inc2 = 1°, α scan2 = 2α scan1 ) provides a better flip angle range (2.4-35°), in practice, φ inc1 = 1° generates a high CSF signal. This can result in an extra signal and a residual artifact in the proximity of the ventricles. To reduce the CSF signal's effect, it was found worthwhile to use φ inc2 = 1.5° (Fig. S4a). As the change in relative variability as a function of TR ( Fig. S4b) is insignificant, the choice of TR can be made by balancing between SAR limitations, on the one hand, and scan duration, on the other hand. A TR of 10 ms provided a practical tradeoff. Our examination of the effect of the R FA on the flip angle range showed that the higher the R FA , the better (Fig. S4c). However, to keep SAR within the "Normal" level, it was found that R FA in the range of 1.6-2 (with TR = 10 ms) provides a suitable flip angle range. In case of adopting "First level" SAR limit, one can increase the range of the flip angles.
Global phase corrections. In practice, the phase ( ∠S ) of the signal S at a voxel is comprised of the steadystate phase θ(α, T 2 , T 1 ) plus a global phase θ 0 . The global phase θ 0 arises from several factors, with a dominant contribution from B 0. It can be eliminated by repeating the scan twice, once with + φ inc and once with -φ inc, and setting θ(α, T 2 , T 1 ) = ∠ S +ϕ inc · conj S −ϕ inc /2 (as was shown in Ref. 27 ). The implemented acquisition thus includes four scans: the two scans (φ inc1, α scan1 ) and (φ inc2, α scan2 ) and their repetition with a negative phase increment to remove θ 0 . Calculating the θ 1 and θ 2 in this method does not result in phase wrapping, since after the global phase removal, the signals' phase is in the range of 0° to ~ 50°. Estimation algorithm. The actual estimation algorithm included two main steps, per voxel, namely the removal of the global phase (θ 0 ) and an estimation of T 2 and α from (θ 1 ,θ 2 ) using linear interpolation. An additional step was established for low flip angles because low flip angles result in (θ 1 ,θ 2 ) measurement pairs close to the edges of the "balloon" (Fig. 2b), a region where interpolation is an ill-posed problem. Low flip angles are relevant for whole-brain imaging because despite the flip angle of the first scan being set to α scan1 = 15° , the actual whole-brain RF field distribution results in a flip angle in the range of ~ 4° to 22° (even reaching below 4° for some regions, see a representing distribution in Fig. 1). Brain regions where very low flip angles (~ 4-6°) are typically reached are the cerebellum, midbrain, and brainstem, as well as some regions in the temporal lobe. The added step to handle low flip angles takes advantage of two aspects: i) that α changes slowly in space, and ii) that for small flip angles (α < 20°) the phase θ is linear with T 2 , and that the slope itself is linear with the flip angle α. Detailed description of this step are in the "Materials and methods" section. Figure S5 shows the improvement attained using the second step for the low flip angles. Additional steps were also performed to improve the estimation for the expected low values of (θ 1 ,θ 2 ), which, due to noise, results in negative values (see "Materials and methods").

T 1 corrections.
As mentioned, phase dependence on T 1 is small, but it can account for ~ 15% of the final T 2 estimation. To reduce the error due to T 1 in human imaging voxels were classified as either "high" or "low" T 1 by empirically thresholding S α scan2 / S α scan1 . Separate maps-T 2 (θ 1 , θ 2 ) and α(θ 1 , θ 2 )-were used for each classification, based on T 1 = 1 s (representing WM) and T 1 = 2 s (the rest). With this correction, the error was further reduced (shown in Fig. S6). A detailed description of the algorithm is provided in the "Materials and methods" section.

Results
To examine the estimation bias and estimation variability we conducted two imaging experiments with phantoms, one with tubes filled with agarose suspension, the other with a 3D head-shaped phantom. In the first experiment ( Fig. 3a), the variability was × 1.4 smaller than with SE-SE (0.5 ms compared to 0.7 ms). The a slope and the relative deviation error (see Eq. 1) calculated between the T 2 from this method and the T 2 from SE-SE were 1.01 and Scientific Reports | (2022) 12:14088 | https://doi.org/10.1038/s41598-022-17607-z www.nature.com/scientificreports/ 0.5%. Thus, the phase-based method provides a small bias and a lower variability compared to SE-SE, while the scan duration is × 2.3 faster.
In the second experiment, a specially designed 3D head-shaped brain-like phantom was used to examine the capability to cope with an RF field distribution similar to that in the brain. The "brain" had a uniform T 2 , which helped to separate the two parameters we sought to estimate, α and T2. Our results show low variability in T 2 (std(T 2 phase-based-method )/std(T 2 SE-SE ) = 0.46) and an RF field map estimation with little bias (a 4% average deviation from the map acquired with the vendor's pulse sequence), see Fig. 3b. Even low flip angles, in the ill-posed area of the "balloon", were well determined using the implemented estimation algorithm (Fig. S8).
The contribution of the B 1 correction to the T 2 estimate can be seen in Fig. 4. It compares T 2 maps extracted from a set of four scans (two pairs) to T 2 maps extracted from a single pair-as in Ref. 27 -using either of the pairs (either pair 1: φ inc1 = 3° and φ inc1 = − 3°, with α scan1 ; or pair 2: φ inc2 = 1.5° and φ inc2 = − 1.5°, with α scan2 ). It can be seen that for both phase increments the RF field inhomogeneity results in either underestimated or overestimated T 2 values, depending on the actual flip angle in each voxel (see Fig. 2a for phase dependence on flip angle). The 4-scans result, which combines both phase increments, provides a uniform T 2 map of the "brain" tissue in the 3D-head shaped phantom, as expected by the design. Figure 4 also shows the estimated T 2 maps, for human imaging, based on either 4-scans or a single scan-pair. Although more challenging to observe, due to the heterogeneous T 2 distribution in the brain and to the very high T 2 values in the CSF regions, it can also be seen that T 2 , estimated from a single pair, is either underestimated or overestimated compared to 4-scans. This can be observed, for example, in regions such as the cerebellum and the temporal lobes. Table 1 summarizes the results by giving sample T 2 values in white matter, grey matter and CSF. For each tissue 2 sampled regions were chosen as shown in Fig. 4-WM1 and WM2 in white matter tissue, GM1 and GM2 in the grey matter tissue and CSF1, CSF2 in the CSF. The table also shows T 2 values reported in Ref. 18 . Note: the CSF values are underestimated with the current method, as further elaborated in the Discussion section.
Continuing with human imaging, Fig. 5 compares the phase-based method with 1.5 mm isotropic voxels to the gold standard SE-SE, for a T 2 mapping comparison, and to the vendor RF mapping, for an RF field mapping comparison. The α map in Fig. 5c was smoothed by 3 × 3 filter to reduce the effect of local CSF signals (see Fig. S13 for original high resolution B 1 map). The RF field map extracted with the phase-based approach shows a distribution similar to the separately acquired vendor map with, however, noticeable deviations in the ventricles, as well as in some of the CSF region. The ratio of the T 2 values and the relative deviation error between the phase-based method and SE-SE is shown in Fig. 6, for the different volunteers. Over all volunteers the T 2 ratio Bottom-estimated T 2 for each tube as a function of T 2 with SE-SE; a slope = 1.01, relative deviation error = 0.5%. The average standard deviation was 0.5 ms for the phase-based method, and 0.7 ms for SE-SE. (b-d) Comparisons using a 3D-head-shaped brain-like phantom. (b) T 2 and α maps estimated by the phase-based method. (c) T 2 map estimated with SE-SE. And (d) α map estimated using the vendor's RF field mapping scan. Two main cross-sections are shown for all cases, Sagittal and Axial. For comparison, the average T 2 and standard deviation was calculated in the same region of interest (marked by a blue contour for the phase-based method and a red contour for SE-SE). The average deviation between the α maps of the phase-based method and of the vendor's RF mapping was calculated to be 0.56° for the Sagittal plane and 0.84° for the Axial plane.  Supplementary Information S4). Finally, high-resolution whole-brain T 2 mapping was performed with the phase-based method, with 1 mm and 0.85 mm isotropic voxels. To acquire whole-brain high-resolution images, × 5.11 acceleration was usedcombining elliptic sampling and × 2 acceleration in both phase encoding directions. Each of the four scans with 1 mm resolution was 1:13 min giving a total scan time of 4:52 min. For 0.85 mm each scan was 1:42 min long and the total scan time was 6:48 min. Figure 7 shows the estimated T 2 maps for the 0.85 mm scan (Fig. S12  Table 1. Table 1. Estimated T 2 in sample regions of white matter, grey matter and CSF (see Fig. 4). www.nature.com/scientificreports/ shows the 1 mm resolution images). To provide even higher robustness following the reduced SNR of the highresolution datasets, we also incorporated denoising based on a DnCNN deep-learning network 33 (provided in MATLAB, The Mathworks, Natick MA, for Gaussian noise removal). This entailed denoising of θ 1 and θ 2 before the estimation of T 2 . The denoising greatly improved the observed details of the cerebellum structure, a region with especially low flip angles (Fig. 7b).

Discussion
The expected rewards of pushing the limits and moving to 7 T MRI are increased spatial resolution and shorter scan durations. Both these features are essential for clinical and research imaging, all the more so for quantitative methods. However, scanning at 7 T also poses new challenges, including high power deposition and severe RF field inhomogeneity. The extended phase-based method shown here delivers high-resolution brain T 2 imaging while overcoming the above challenges. This is achieved by relying on a modified 3D SGRE sequence, using the phase of the signal to encode the T 2 dependence. The 3D SGRE images are also highly robust to B 0 inhomogeneity. This can be seen in the magnitude images of both the phantom example (Fig. 3a) and the human images (Fig. 5). The SE-SE is more distorted both at the edges of the agar tubes and near the nasal areas in the human images. The B 0 -dependent phase is reliably canceled out by the two scans with opposite phase increments (φ inc ) of the RF pulse train. However, shifts in the global phase between scans may occur, which will require corrections. Similarly, the scans may be sensitive to movements, which will affect the phase. Incorporating a second echo acquisition could be used to correct for both the phase shifts and motion 34 . Aiming to shorten the total scan duration, one can also consider estimation of the global phase from a single pair, thus reducing the number of scans to three. However, in this case careful analysis and phase unwrapping will be required in the third, non-paired, scan. In this case, phase unwrapping can be especially challenging in regions with short T 2 * , where B 0 changes rapidly, resulting in high local changes in the background phase. Very short T 2 * may also affect T 2 estimation due to limited SNR in such regions.
The current implementation used a non-selective hard pulse for the 3D acquisition. Although this works well for whole brain acquisition as in this study, in other cases it can be a limitation. For faster acquisition and to limit potential aliasing, the use of slab-selective pulses is beneficial. Figure S9 shows that as long as the slab is thick enough, compared to the slice thickness, the estimated T 2 is correctly estimated. However, for a single slice-selective acquisition, the simulation by which the T 2 and RF field maps are estimated must also account for the slice profile. This was already demonstrated in other T 2 mapping methods such as balanced SSFP 19 .
Another sensitivity of the method that requires discussion is the sensitivity to movement and potential inaccuracy in the RF pulse phase. Although we did not observe noticeable movement in our human scanning, a simulation to examine these vulnerabilities was performed (see Supplementary Information, Section S5). The movement was simulated assuming a constant velocity during the scan, which will result in an additional parabolic phase term accumulated during the scan. Examining the error due to potential head movement of 1-2 voxels during the scan, it resulted in a small error, less than 1% for a movement of up to 5 mm/min. However, www.nature.com/scientificreports/ for large movement within a voxel, such as due to flow, the error of the estimated T 2 can be significant; reaching 20%, for a velocity of 0.5 mm/s. Two simulations were also performed to analyze possible hardware inaccuracies: (i) a constant error in the actual RF-phase increment, (ii) a randomly distributed error in the actual phase of the RF pulse. In the first case, a constant error of 0.1° resulted in < 4% error, In the second, a randomly distributed error with σ = 0.2° resulted in a negligible error with standard deviation of 0.07 ms in the estimated T 2 . It is also important to note that the estimation of the T 2 in the CSF and other tissues with high T 2 values (> 0.5 s) is challenging with this method, since the signal's phase curve slowly converges for T 2 > 100 ms (see Fig. 2a) and so the T 2 contours in the (θ 1 ,θ 2 ) space grow denser with T 2 (see Fig. 2b). In addition, local intensity drops in CSF voxels, resulting in low SNR voxels, can occur due to fluid movement (purple arrows in Fig. 5 point to such area), thus further limiting T 2 estimation of CSF.
The important advantage of the phase-based approach for T 2 mapping is its whole-brain coverage ability. The method shows robust results in the brainstem region and even in parts of the spinal cord (see Fig. 5). These results are achieved without the need for additional hardware to reduce the RF field inhomogeneity, such as dielectric pads or multi-channel transmit coil. Naturally, the method can also benefit from a dielectric pad or multi-channel transmit coils to improve the SNR, especially in regions with low flip angles. The current configuration (φ inc1 , φ inc2 , α scan1 , R FA , and TR) was designed for the RF field distribution in the brain, and was shown to robustly extract the RF field distribution in the 3D head-shaped phantom (which has a slightly larger RF field inhomogeneity than in vivo). If another region will be of interest, the configuration-the RF pulse phase increments and the scan flip angles-can be adapted accordingly.
It is worth noting that θ 1 on its own, calculated from the first pair of scans (with φ inc1 = ± 3°), achieves a "T 2 weighted" image (see Fig. S10 for the 0.85 mm case), unlike the magnitude of these scans. θ 1 , however, suffers from pronounced RF field inhomogeneity, which is removed by using two sets of scans (giving θ 1 and θ 2 ), as was implemented here, allowing the generation of T 2 maps.
In our study, the estimation algorithm is based on an interpolation procedure, where the simulated data serves as the ground-truth. This method is similar to the dictionary-based approach in MRF, but is based on two measurement points (θ 1 , θ 2 ) that allow us to represent the parameters of interest, T 2 and α, in the (θ 1 , θ 2 ) 2D space. This offers the advantage of mapping the T 2 of interest by a simple linear interpolation. An improvement in the estimation algorithm was implemented in the low flip angles' range, which extended the viable flip angles (Figs. S5 and S8). In this study, we demonstrated the low variability and small bias of the estimations in both simulations and phantom experiments. In the phantom experiment with agar tubes, the method provides T 2 estimation with low variability-a × 3.2 (1.4 × 2.3) lower variability-to-scan-time factor than that of SE-SE. The T 2 values were estimated by the phase-based method with a small bias (a slope = 1.01 and relative deviation error of 0.5% compared to SE-SE).
However, the in-vivo T 2 ratio of the phase-based method to SE-SE was 0.79 ± 0.16 for WM and 0.86 ± 0.19 for GM. Similarly, there is a ratio of × 0.82 and × 0.88 between the reported values with 4-scans in Table 1 to the values in Ref. 18 . This result is also similar to the results in Ref. 22,34 . Possible reasons for the different ratios found for WM and GM are a partial volume of GM and CSF as well as deviations due to T 1 . Although T 1 has a small impact on the phase of the signal and its effect was reduced in our implementation. The ~ 0.8 ratio between the T 2 estimated by the phase-based approach and by SE-SE could arise for several reasons, among which are a contribution due to exchange and magnetization transfer 35 , diffusion 36 , and different contributions of the fast and slow T 2 components to the two methods 37,38 . For the magnetization transfer no discrepancy was observed between the estimated T 2 values in the agarose tubes, although exchange mechanisms are known to be at work in agarose and therefore produce magnetization transfer effects. However, different effects of exchange in the living tissue can still be a factor contributing to the acquired complex signal of the steady state acquisition. We also examined potential diffusion contributions to the estimated T 2 by scanning a sample of smoked fish (which had and ADC of ~ 0.6 × 10 −3 mm 2 /s, similar to white matter) and did not observe a significant effect (not shown). One of the potential factors is the larger contribution of the fast-relaxing species compared to SE-SE, primarily due to much shorter echo times, which was also observed in several previous studies 38 . Thus, although the estimated T 2 was robustly repeated in the volunteers' data, the resulting ratio between the phase-based method and SE-SE in-vivo still requires further analysis.
The fast high-resolution T 2 maps of the whole brain that were acquired-1 mm isotropic in 4:52 min and 0.85 mm isotropic in 6:49 min-offer a significant clinical gain. Further acceleration of the method should be possible. One option is to reduce the TR, however, this will require switching the SAR monitoring to the less restrictive "First" level. For this, the effect of the TR on the variability of the estimation was examined (see Fig. S4) and showed that shorter TR result in similar estimation variability. In addition, acceleration methods, tuned to the 3D SGRE acquisition and employing the Compressed Sensing technique, can achieve even higher acceleration factors. We also demonstrated the option of employing denoising based on deep-learning techniques that is trained to remove Gaussian noise. This further improves the quality of the images and can be used to further accelerate the scan.
Overall, the extension of the phase-based steady-state method to estimate both T 2 and RF field map, demonstrated in this work, provides a fast and high-resolution acquisition method for quantitative T 2 mapping of the whole brain at 7 T acquired with a single-channel transmit coil. Standardized high-resolution methods are imperative for 7 T MRI to advance multi-site studies and promote personalized medicine.

Materials and methods
Bloch simulations. 1D single voxel simulations based on the Bloch equations were performed with a custom MATLAB (The Mathworks, Natick MA) code 39 to examine the signal in steady state. The simulations included an excitation pulse, an acquisition and a net total spoiler (including the area of the acquisition) of 3/Δx (Δx the 1D voxel size). The number of initial repetitions to reach steady-state ("dummy scans") was set to 500, which was verified to provide reliable steady states. Following the dummy scans, a single acquisition was simulated. The simulation was repeated over a grid of flip angles and T 2 values, for different values of T 1 , φ inc , and TR. The grid covered T 2 from 0 to 200 ms with a resolution of 4 ms, and flip angles from 0° to 70° with a resolution of 1°. The resulting θ(T 2 , α) map was interpolated prior to its use in the estimation algorithm with 1 ms in T 2 and 0.1° in alpha, generating θ 1 (T 2 , α) and θ 2 (T 2 , α) for relevant φ inc and R FA factors.
Estimation algorithm. The estimation algorithm included the following steps: Preparatory step #0.1: global phase removal. In practice, the phase ( ∠S ) of the signal S at a voxel is comprised of the steady-state phase θ(α, T 2 , T 1 ) plus a global phase θ 0 . The global phase θ 0 arises from several factors, with a dominant contribution from B 0. It can be eliminated by repeating the scan twice, once with + φ inc and once with -φ inc, and setting θ(α,T 2 ,T 1 )=∠ S +ϕ inc · conj S −ϕ inc /2 (as was shown in Ref. 27 ). The implemented acquisition thus includes four scans: the two scans (φ inc1, α scan1 ) and (φ inc2, α scan2 ), and their repetition with a negative phase increment to remove θ 0 .
Preparatory step #0.2 (optional): denoising. For high-resolution human imaging, a denoising procedure based on a DnCNN deep-learning network 33 (provided in MATLAB 2021a, for Gaussian noise removal) was incorporated. The denoising procedure was implemented on the measured θ1, θ2 with the command denoised_ θ1 = denoiseImage(θ1, net), where the net was set by the command net = denoisingNetwork(' dncnn').
As mentioned, phase dependence on T 1 is small, but it can account for ~ 15% of the final T 2 estimation. Thus, in human imaging, to reduce the error due to T 1 , voxels were classified as either "high" or "low" T 1 by empirically thresholding S α scan2 / S α scan1 . Separate maps-T 2 (θ 1 , θ 2 ) and α(θ 1 , θ 2 )-were used for each classification, based on T 1 = 1 s (representing white matter-WM) and T 1 = 2 s (the rest). With this correction, the error was further reduced (see simulation results in Fig. S6).
Estimation step #2: T 2 estimation update for low flip angles. First, the flip angles α found in the previous step were smoothed, generating α smoothed . For low flip angle voxels with α smoothed < 4.5°, the flip angles were temporarily set to α temp = 4.5°, and the matching temporary T 2 quantities, T 2-temp , were found by interpolation-using α temp and θ 2 (the phase from the scan using the higher flip angle, αscan 2 = RFA•αscan 1 ). The final T 2 was found through the linear connection T 2 = (α temp /α smoothed )•T 2-temp .
Validation of the estimation algorithm was performed by generating N = 100 noisy repetitions of each point in the simulated datasets of θ 1 (T 2 ,α) and θ 2 (T 2 ,α). This was done using a fixed noise which resulted in the SNR varying with T 2 and α, depending on the intensity at each point. The noise was fixed to produce an SNR of 180 for the simulated data at T 2 = 38 ms and α = 13°; resembling the SNR in the human images acquired with 1.5 mm resolution. The SNR was set as an average SNR over the two signals |S 1 | and |S 2 |. To validate the simulations, the standard deviation of T 2 was compared to a measured one in an agar-tubes experiment, both with the same SNR. For this validation two agar-tubes were used-with T 2 values of T 2 = 34 ms and T 2 = 38 ms, representing WM and GM at 7 T. The flip angle distribution in this experiment was uniform (α = 13°). The measured and simulated SNR was 298, resulting in a T 2 standard deviation of 0.36 ms in the measurement and 0.32 ms in the simulation, providing comparable results. The variability and bias of the method, under the simulated noise, were examined as a function of T 2 , α, φ inc , TR and R FA .
Pulse sequence considerations. The sequence is based on a Siemens 3D GRE sequence that was modified to enable control over both the φ inc and the gradient spoiler moment. The RF pulse we used was a hard pulse.
An important aspect to consider is the gradient spoiler moment intensity and its effect on the T 2 estimation, as well as on image artifacts (in the form of residual signals from spurious echoes). A set of scans was performed to examine the spoiler effect. The gradient spoiler moment needs to provide complete dephasing inside a voxel, which defines a preferable gradient moment size to be 1/Δr (Δr = √(Δx 2 + Δy 2 + Δz 2 )). We found it useful to add a parameter to the pulse sequence that directly controls the net gradient spoiler moment (after all previous gradients had been rephased). The net spoiler was set to be equally distributed in all three directions, which was found useful in reducing artifacts. However, our experiments also showed that the gradient moment affected the measured phase, and thus the estimated T 2 . Figure S7a shows  www.nature.com/scientificreports/ provide dephasing for Δr 0.9 mm. As shown in Fig. S7b, under this moment, the estimated T 2 did not change for the voxel sizes tested.
MRI scanning. All scans in this study were performed on a 7 T MRI system (MAGNETOM Terra, Siemens Healthcare, Erlangen) using a commercial 1Tx/32Rx head coil (Nova Medical, Wilmington, MA). When comparing the results of the phase-based method to SE-SE, inside a region, the relative deviation error from the fit was calculated as where a slope is the slope found for each fit, and N is the number of voxels in the comparison.
Phantom imaging. Five tubes with agar concentrations of 1.5, 2, 2.5, 3 and 3.5% were used to compare the phase-based T 2 estimation to the gold standard SE-SE, using three TE values (10, 30 and 50 ms). A 3D headshaped phantom that was designed to model the RF field distribution in the brain was used to examine the T 2 and RF field estimation. This phantom was originally designed to include three sub-compartments 30 , suitable for mimicking brain, muscle and lipid tissues. However, the version used in this study was filled with two "tissue" types: the inner compartment mimicked the "brain" and the outer one, "muscle" (the planned lipid layer was also filled with "muscle"). Both compartments contained 0.1 mM gadopentetate dimeglumine (GdDTPA), for a T 1 close to that of human white matter, and consisted of an agarose suspension of 2.5% and 3% for the "brain" and "muscle" compartments, respectively. NaCl (5.5 gr/L) was used to achieve an in-vivo-like RF field distribution. For details, see Ref. 30 .
α maps from the phase-based method were compared to the equivalent α maps generated by the vendor. As the RF field maps provided by the vendor are scaled to 90°, they were rescaled to the α scan of the phase-based method, before comparison. The average deviation between the α maps by the phase-based method and by the vendor were calculated in two main planes (Sagittal and Axial).
Human imaging. All methods were carried out in accordance with the Weizmann Institute of Science guidelines and regulations. This study was approved by the Internal Review Board of the Wolfson Medical Center (Holon, Israel) and all scans were performed after obtaining informed suitable written consents. Human scanning of six volunteers with isotropic 1.5 mm resolution was acquired for the comparison with SE-SE. The comparison was performed after the SE-SE and the phase-based method images were realigned using SPM12 (https:// www. fil. ion. ucl. ac. uk/ spm/) to ensure there was no movement between the scans.
An additional volunteer was scanned with the phase-based method with 1 mm and 0.85 mm resolution. These scans were acquired with an acceleration of × 5.11-using elliptical sampling and × 2 acceleration in both phase encoding directions. The BART 40 software was used to reconstruct this dataset.

Data availability
All scans collected in this study were performed according to procedures approved by the Internal Review Board of the Wolfson Medical Center (Holon, Israel). Since this protocol was not defined as an open repository, the data is not provided, to provide the ethics and privacy issues of clinical data. The code will be made available via a request to the corresponding author.